Distribution modulo one
In this chapter, we give a brief account of the theory of uniform distribution modulo one. For complements, and for additional bibliographic references, the reader is directed to the monographs [232, 411].
1.1 Weyl’s criterion
In the years 1909−1910 there appeared three papers by Bohl , Sier-piski  and Weyl  devoted partly to the distribution of real sequences. A few years later, Weyl [732, 733] developed the results of these papers, giving birth to the study of uniform distribution.
DEFINITION 1.1. The sequence (xn)n≥1 of real numbers is dense modulo one if every interval of positive length included in ...